Example. A jacket is reduced by $20\%$ and then increased by $20\%$. Its original price is 80.

Use multipliers: $80\times 0.80\times 1.20=76.80$. Equal percentage decreases and increases do not cancel.

A unit fraction has numerator 1, e.g. $\frac{1}{4}$, $\frac{1}{5}$, $\frac{1}{10}$. Dividing by a whole number $n$ gives exactly the same result as multiplying by the unit fraction $\frac{1}{n}$ — its reciprocal, or multiplicative inverse — because $n\times \frac{1}{n}=1$.

Example: $3÷5$ and $3\times \frac{1}{5}$ both give $\frac{3}{5}$, because sharing into 5 equal parts is the same as taking one of those fifths, three times over.

This is the concept behind "keep, change, flip": dividing by any fraction $\frac{c}{d}$ means multiplying by its multiplicative inverse $\frac{d}{c}$, since $\frac{c}{d}\times \frac{d}{c}=1$.

Always simplify fractions to their lowest terms. Divide numerator and denominator by their HCF. For example, $\frac{12}{18}=\frac{2}{3}$ (dividing both by 6).

To express $A$ as a fraction of $B$, write $\frac{A}{B}$ and simplify. The result can be less than 1 or greater than 1, depending on whether $A$ is smaller or larger than $B$.

(a) In a class of 40 students, 18 are boys. Express the number of boys as a fraction of the whole class. (b) A jug holds 250 ml; a bottle holds 400 ml. Express the bottle's capacity as a fraction of the jug's.

This uses the same ratio as "express as a percentage" (see Percentages) — just left as a simplified fraction instead of multiplied by 100%.

Divide the numerator by the denominator:

If the division terminates, it's a terminating decimal. If it repeats, it's a recurring decimal.

A fraction $\frac{a}{b}$ in its simplest form gives a terminating decimal if and only if the denominator $b$ has no prime factors other than 2 and 5. Otherwise, the decimal recurs.

Fraction → decimal: divide numerator by denominator.
Decimal → percentage: multiply by 100.
Percentage → decimal: divide by 100.
Fraction → percentage: divide numerator by denominator, then multiply by 100.

Memory trick — the D/P slide: decimal and percentage differ by only a 2-place slide of the decimal point. Slide it right two places for Decimal → Percentage; slide it left two places for Percentage → Decimal.

Learn these by heart — they save precious time in the exam. Many percentage and probability questions rely on quick recall of these facts.

A dot is placed above the first and last digits of the repeating block:

If only one digit repeats, use one dot: 0.$\dot{3}$
If a block repeats, put a dot on the first and last digit of the block: 0.$\dot{1}$ $\dot{7}$
Digits between the dots are included in the repeating block: 0.$\dot{1}4285\dot{7}$ repeats 142857.

Make sure you multiply by the right power of 10. Count only the repeating digits, not the non-repeating ones. When there's a non-repeating part before the recurring block, you need two multiplications (see Example 3).

Question. Without a calculator, arrange these in ascending order: $\frac{7}{20}$, $0.36$, $34\%$, $0.\dot{3}\dot{5}$.

The trap is treating $\frac{7}{20}$ and $0.\dot{3}\dot{5}$ as the same number — one terminates at $0.35$, the other never stops, so precision matters.

Question. Convert $0.2\dot{4}\dot{5}=0.2454545\dots$ to a fraction in its lowest terms.

Question. A recipe states that flour must make up $0.375$ of the total mass of dry ingredients. (a) Write $0.375$ as a fraction in its lowest terms and as a percentage. (b) If the dry ingredients weigh $960\text{g}$ in total, what mass of flour is needed?

To find $x$% of a quantity Q, calculate:

$\frac{x}{100}\times Q$

To express A as a percentage of B:

$\frac{A}{B}\times 100$%

Always check which quantity is the "whole" (denominator). A common mistake is dividing by the wrong number.

Always divide by the original value, not the new value. The "original" is the starting amount before the change.

Compound interest $earned=A-P$

After 3 years at 5%:

The difference grows larger with more years and higher rates.

For depreciation (value going down), use $A=P(1-\frac{r}{100}{)}^n$. The multiplier is less than 1.

The final price represents 100% + 20% 120% of the original.

The sale price represents 100% - 15% 85% of the original.

Do NOT just calculate 20% of 660 and subtract. That would give $660-132=528$, which is wrong. You must divide by the multiplier.

A percentage is just a fraction out of 100, which is also a decimal:

Write fractions in simplest form before converting. If $\frac{40}{100}$ is needed as a fraction, give $\frac{2}{5}$.

A recurring decimal repeats a block of digits forever. We show this with a dot over each repeating digit (or a bar over the block):

All recurring decimals can be written as exact fractions.

To convert a recurring decimal: multiply by $10^n$ where $n$ is the length of the recurring block, then subtract the original to eliminate the recurring part.

Total amount = $P+I$

The interest is the same every year because it is always calculated on the original principal. This makes it easy to calculate but means your money grows in a straight line, not exponentially.

Question. An investment grows from £2000 to £2205 after $2$ years of compound interest, with no extra deposits. Find the annual interest rate.

When the rate is the unknown rather than the final amount, isolate the power first and undo it with a root — a square root for 2 years, a cube root for 3 years, and so on.

Question. A jacket's price is first increased by $20\%$ ahead of a preview sale, then reduced by $25\%$ in the final clearance. The original price was $50. (a) Find the final price. (b) Find the overall percentage change from the original price.

This is the misconception from the tips above made concrete: the $25\%$ is taken off the increased price ($60),nottheoriginal($50), so the two changes never cancel exactly.

Question. Sam invests $3000 for $4$ years. Bank A offers $6\%$ per year simple interest. Bank B offers $5\%$ per year compound interest. Which bank gives the better return, and by how much (to the nearest cent)?

Don't assume compound interest always wins — over a short time at a lower rate, a generous simple rate can still come out ahead.

Question. A coat costs $\pounds 68$ after a $15\%$ reduction. Find the original price.

Marking. 1 mark per point. Never add $15\%$ back onto $\pounds 68$ — that gives $\pounds 78.20$, which is wrong. Divide by the multiplier.

Question. An investment of $\pounds 2500$ earns $4\%$ compound interest per year. Find its value after $3$ years, and the total interest earned.

Marking. 1 mark per point. Compound interest is not $3\times 4\%=12\%$ — that would give $\pounds 300$. Each year's interest earns interest too.

Question. Work out $2\frac{1}{4}÷\frac{3}{8}$, giving your answer as a mixed number.

Marking. 1 mark per point. Always convert mixed numbers first — dividing the whole parts and fraction parts separately does not work.

Reasoning prompt. A price changes by a percentage and then changes again. Which representation makes repeated change safest?

For a percentage increase, the multiplier is greater than one; for a decrease, it is less than one.

After a 15% discount, a jacket costs £68. What was its original price?

The sale price is 85% of the original, so $0.85P=68$.

$P=68÷0.85=\boxed{80}$.

Check: 15% of 80 is 12, and $80-12=68$.

Let $x=0.1\dot{6}=0.1666\dots$. Then $10x=1.6666\dots$ and $10x-x=1.5$, so $9x=\frac{3}{2}$ and $x=\frac{1}{6}$. Multiplication must shift one complete repeating block before subtraction.